Characteristic property of the subspace topology

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Statement

Diagram
Let (X,J) be a topological space and let (S,JS) be any subspace of (X,J)[Note 1]. The characteristic property of the subspace topology[1] is that:
  • Given any topological space (Y,K) and any map f:YS we have:
    • (f:YS is continuous)(iSf:YX is continuous)

Where iS:SX given by iS:ss is the canonical injection of the subspace topology (which is itself continuous)[Note 2]

Proof

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See also


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Notes

  1. Jump up This means SP(X), or SX of course
  2. Jump up This leads to two ways to prove the statement:
    1. If we show iS:SX is continuous, then we can use the composition of continuous maps is continuous to show if f continuous then so is iSf
    2. We can show the property the "long way" and then show iS:SX is continuous as a corollary

References

  1. Jump up Introduction to Topological Manifolds - John M. Lee